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In [[order theory]] and [[functional analysis]], a subset ''S'' of a [[vector lattice]] is said to be '''solid''' and is called an '''ideal''' if for all ''s'' in ''S'' and ''x'' in ''X'', if |''x''| ≤ |''s''| then ''x'' belongs to ''S''.
An [[ordered vector space]] whose order is Archimedean is said to be ''Archimedean ordered''.
If ''S'' is a subset of ''X'' then the '''ideal generated by ''S''''' is the smallest ideal in ''X'' containing ''S''.
An ideal generated by a singleton set is called a '''principal ideal''' in ''X''.
== Examples ==
* The intersection of an arbitrary collection of ideals is an ideal.
== Properties ==
* A solid subspace of a vector lattice ''X'' is necessarily a sublattice of ''X''.
* If ''N'' is a solid subspace of a vector lattice ''X'' then the quotient ''X''/''N'' is a vector lattice (under the canonical order).
== See also ==
* [[Vector lattice]]
== References ==
* <!-- -->
<!--- Categories --->
[[Category:Functional analysis]]
An [[ordered vector space]] whose order is Archimedean is said to be ''Archimedean ordered''.
If ''S'' is a subset of ''X'' then the '''ideal generated by ''S''''' is the smallest ideal in ''X'' containing ''S''.
An ideal generated by a singleton set is called a '''principal ideal''' in ''X''.
== Examples ==
* The intersection of an arbitrary collection of ideals is an ideal.
== Properties ==
* A solid subspace of a vector lattice ''X'' is necessarily a sublattice of ''X''.
* If ''N'' is a solid subspace of a vector lattice ''X'' then the quotient ''X''/''N'' is a vector lattice (under the canonical order).
== See also ==
* [[Vector lattice]]
== References ==
* <!-- -->
<!--- Categories --->
[[Category:Functional analysis]]
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